Question: Determine how many solutions exist for the system of equations. ${12x+3y = -27}$ ${-12x-3y = -3}$
Convert both equations to slope-intercept form: ${12x+3y = -27}$ $12x{-12x} + 3y = -27{-12x}$ $3y = -27-12x$ $y = -9-4x$ ${y = -4x-9}$ ${-12x-3y = -3}$ $-12x{+12x} - 3y = -3{+12x}$ $-3y = -3+12x$ $y = 1-4x$ ${y = -4x+1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x-9}$ ${y = -4x+1}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.